Control system design for a mixing system with multiple inputs

ABSTRACT

A control system for mixing at least two materials in a physical system having two or more tanks comprises at least two actuators, each actuator being operable to introduce a material into a first tank to form a first mixture, the first mixture flowing into a second tank to form a second mixture and a controller operable, based on a commanded input, to control the at least two actuators to obtain a density of either the first mixture or the second mixture and a volume flow rate of the second mixture out of the second tank, wherein the density is controlled independently from the volume flow rate.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit under 35 U.S.C. § 119(e) ofU.S. Provisional Application Ser. No. 60/671,392 filed Apr. 14, 2005 andentitled “Implementation of Alternative Cement Mixing Control Schemes,”by Jason D. Dykstra, et al, which is incorporated herein by referencefor all purposes.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

REFERENCE TO A MICROFICHE APPENDIX

Not applicable.

FIELD OF THE INVENTION

The present disclosure is directed to a control system design for amixing system having multiple inputs, and more particularly, but not byway of limitation, to a control system design for a cement mixing systemused in well bore servicing applications.

BACKGROUND OF THE INVENTION

A control system typically comprises one or more physical systemcomponents under some form of automated control that cooperate toachieve a set of common objectives. The control system may be designedto reliably control the physical system components in the presence ofexternal disturbances, variations among physical components due tomanufacturing tolerances, and changes in commanded input values forcontrolled output values, such as a cement mixture density, for example.The control system may also be designed to remain stable and avoidoscillations within a range of specific operating conditions.

In a well bore environment, a control system may be used when mixingmaterials to achieve a desired mixture output. For example, whendrilling an oil or gas well, it is common to install a tubular casinginto the well bore and cement the casing in place against the well borewall. A cement mixing system that supports well bore servicingoperations, such as cementing casing into a well bore, may be designedwith a control system configured to provide a desired volumetric flowrate of mixed cement having a desired density. In particular, the cementmixing control system may control valves that allow the in-flow of drycement material and water to obtain the desired cement mixture densityand desired cement mixture volumetric flow rate. The control system mayoperate, for example, by monitoring the cement mixture flow rate anddensity, and by regulating an in-flow water valve and an in-flow drycement material valve. However, because such systems conventionallycontrol the output parameters, such as cement mixture flow rate anddensity, dependently, these systems tend to have long lag times in theresponse of one valve to changes in the position of the other valve.This can lead to unacceptable oscillations in the monitored parameters,and difficulty in stabilizing the system. Therefore, to make the systemmore stable, it would be desirable to control output parameters, such asa mixture flow rate and a mixture density, for example, independently ofeach other. Accordingly, a need exists for a mixing control system withmultiple inputs that decouples the effects of changes in the commandedoutputs.

SUMMARY OF THE INVENTION

Disclosed herein is a control system for mixing at least two materialsin a physical system having two or more tanks comprising at least twoactuators, each actuator being operable to introduce a material into afirst tank to form a first mixture, the first mixture flowing into asecond tank to form a second mixture and a controller operable, based ona commanded input, to control the at least two actuators to obtain adensity of either the first mixture or the second mixture and a volumeflow rate of the second mixture out of the second tank, wherein thedensity is controlled independently from the volume flow rate.

Further disclosed herein is a control system for mixing at least twomaterials in a tank comprising at least two actuators, each actuatorbeing operable to introduce a material into the tank to form a mixtureand a controller operable, based on a commanded input, to control the atleast two actuators to obtain a density of the mixture and a volume flowrate of the mixture out of the tank, wherein the density is controlledindependently from the volume flow rate.

Further disclosed herein is a control system for mixing at least twomaterials comprising a controller operable to control a physical systemcomprising a plurality of actuators, each actuator being operable tointroduce a material into a first tank to form a first mixture, whereinthe controller operates the actuators to control a desiredcharacteristic of the first mixture.

Further disclosed herein is a method for mixing at least two materialscomprising introducing the at least two materials into a physicalsystem, combining the at least two materials to form a mixture thereof,and independently controlling a desired characteristic of the mixture ofthe at least two materials and a desired parameter of the physicalsystem.

Further disclosed herein is a method for mixing at least two materialscomprising determining a commanded combined mass flow rate of the atleast two materials into a first tank [commanded dm_(in)/dt],determining a commanded combined volume flow rate of the at least twomaterials into the first tank [commanded dv_(in)/dt], and independentlycontrolling, based on the commanded dm_(in)/dt and the commandeddv_(in)/dt, a density of a mixture of the at least two materials in thefirst tank [ρ₁₂]and a volume flow rate of the mixture out of the firsttank [dv₁₂/dt].

These and other features and advantages will be more clearly understoodfrom the following detailed description taken in conjunction with theaccompanying drawings and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present disclosure and theadvantages thereof, reference is now made to the following briefdescription, taken in connection with the accompanying drawings anddetailed description, wherein like reference numerals represent likeparts.

FIG. 1 is a diagram of one embodiment of a physical plant within which acontrol system for a mixing system may be implemented;

FIG. 2 is a block diagram of one embodiment of a control system;

FIG. 3 is a block diagram of one embodiment of a flow modulatorcomponent of the control system of FIG. 2;

FIG. 4 is a block diagram of one embodiment of a flow regulatorcomponent of the control system of FIG. 2;

FIG. 5A is a block diagram of two embodiments of a state feedbackcontroller with command feed forward component of the control system ofFIG. 2;

FIG. 5B is a block diagram of the first embodiment of the state feedbackcontroller with command feed forward component of FIG. 5A;

FIG. 5C is a block diagram of the second embodiment of the statefeedback controller with command feed forward component of FIG. 5A;

FIG. 6 is a block diagram of the components of the control system ofFIG. 2 in the context of the physical plant;

FIG. 7 is a flow diagram illustrating an exemplary use of the controlsystem of FIG. 2;

FIG. 8A is a block diagram of one embodiment of a height observercomponent;

FIG. 8B is a block diagram of one embodiment of a control systemcomprising first and second height observer components;

FIG. 9A is a block diagram of one embodiment of a density observercomponent;

FIG. 9B is a block diagram of one embodiment of a control systemcomprising first and second density observer components;

FIG. 10A is a block diagram of a first portion of an embodiment of acontrol system for a physical plant having a single tank, the firstportion including a height controller;

FIG. 10B is a block diagram of a second portion of the embodiment of thecontrol system for a physical plant having a single tank, the secondportion including a density controller;

FIG. 11 is a block diagram of one embodiment of a control system havinga density observer associated with a first tank and no automated heightcontrol;

FIG. 12 is a block diagram of one embodiment of a control system havinga density observer associated with the first tank and a height observerassociated with a second tank; and

FIG. 13 illustrates one example of a general purpose computer systemsuitable for implementing the several embodiments of the control systemand its various components.

DETAILED DESCRIPTION

It should be understood at the outset that the present disclosuredescribes various implementations of different embodiments of a controlsystem having one or more inputs. However, the present control systemmay also be implemented using any number of other techniques, whethercurrently known or in existence. The present disclosure should in no waybe limited to the descriptions, drawings, and techniques illustratedbelow, including the design and implementation illustrated and describedherein.

In an embodiment shown in FIG. 1, a physical plant 10 to be controlledcomprises a first tank 12 joined by a weir 14 to a second tank 16, afirst actuator 18 dispensing a first material to be mixed, a secondactuator 20 dispensing a second material to be mixed, and an outflowpump 22. A first mixture 13 is formed from the in-flow of the firstmaterial from the first actuator 18 and the second material from thesecond actuator 20 into the first tank 12. When the first mixture 13fills the first tank 12 to the height of the weir 14, the first mixture13 overflows into the second tank 16. The mixture in the second tank 16is referred to as a second mixture 15. A first stirrer 24 and a secondstirrer 26 may be provided to promote homogeneity of the first mixture13 in the first tank 12 and the second mixture 15 in the second tank 16,respectively. The second mixture 15 exits the second tank 16 via anoutflow pump 22 and a discharge line 32. In other embodiments,additional actuators may feed additional materials into the first tank12. In other embodiments, a single tank or three or more tanks may beused.

In an embodiment, the physical plant 10 is a well bore servicing fluidmixing system, such as, for example, a cement mixer used to provide acontinuous stream of a cement slurry for cementing a tubular casingagainst a well bore wall. In this embodiment, a dry cement material maybe fluidized by the introduction of pressurized air, which promotesfluid flow of the dry cement through a first feed line 28, to bedispensed into the first tank 12 through the first actuator 18, forexample, and a carrier fluid, such as water, for example, may flowthrough a second feed line 30 to be dispensed into the first tank 12through the second actuator 20. In this embodiment, the first and secondactuators 18, 20 may be valves, for example. These two materials, thedry cement material and the carrier fluid, are mixed by the firststirrer 24 to obtain the first mixture 13. Non-fluidized sand or otherparticulate matter may be dispensed through a third actuator (notshown), such as a screw feeder, for example, into the first tank 12 tomix with the cement slurry. In various embodiments, any of the actuatorsmay comprise a valve, a screw feeder, an augur, an elevator, or othertype of actuator known to those skilled in the art. The first mixture 13is preferably substantially homogenous. The physical plant 10 shouldprovide the cement slurry via the outflow pump 22 and the discharge line32 at a volumetric flow rate sufficient to support the well boreservicing operation, and should mix the dry concrete material, thecarrier fluid, and any particulate material in appropriate proportionsso that the cement slurry dispensed thereby achieves a desired density.The physical plant 10 in other embodiments may support other mixingoperations of other materials. For example, in another embodiment,proppants and a carrier fluid may be dispensed through the first andsecond actuators 18, 20 into the first tank 10 to form a portion of afracturing fluid.

FIG. 1 also identifies several parameters of a control system, e.g., afirst control system 100 depicted in FIG. 2, coupled to the physicalplant 10 and functional to control the operation thereof. The first tank12 has a cross-sectional area represented by the constant A₁, and thesecond tank 16 has a cross-sectional area represented by the constantA₂. The height of the first mixture 13 in the first tank 12 isrepresented by the variable h₁, and the height of the second mixture 15in the second tank 16 is represented by the variable h₂. The volumetricflow rate of the first material, for example dry cement, through thefirst actuator 18 into the first tank 12 is represented by dv₁/dt. Thevolumetric flow rate of the second material, for example water, throughthe second actuator 20 into the first tank 12 is represented by dv₂/dt.The volumetric flow rate of the first mixture 13, for example a cementslurry, over the weir 14 into the second tank 16 is represented bydv₁₂/dt. The volumetric flow rate of the second mixture 15 out of thesecond tank 16 and through the outflow pump 22 is represented bydv_(s)/dt.

The first control system 100 disclosed hereinafter is expected to reducecontrol oscillations due to system time lags and to promote independentcontrol of a mixture density and a mixture flow rate.

Turning now to FIG. 2, a first control system 100 is depicted that iscoupled to the physical plant 10 of FIG. 1 and controls the firstactuator 18 and the second actuator 20. In an embodiment, the firstcontrol system 100 controls the actuators 18, 20 to obtain sensedparameter values that approach or equal the following input parametervalues that are input into the first control system 100 by an operatorthrough an interface with the first control system 100: a height h₂ ofthe second mixture 15 in the second tank 16, a volumetric flow ratedv_(s)/dt of the second mixture 15 out of the second tank 16, and adensity ρ_(s) of the second mixture 15 out of the second tank 16. Inother embodiments, the first control system 100 may control the firstactuator 18 and the second actuator 20 to obtain sensed parameter valuesthat approach or equal other input parameter values. For example, thefirst control system 100 may be said to control the volumetric flow rateof the first mixture 13 over the weir 14 into the second tank 16,represented by dv₁₂/dt, and the density of the second mixture 15 as itleaves the second tank 16, represented by ρ_(s). However, the firstcontrol system 100 actually controls the first actuator 18 and thesecond actuator 20, for example, by adjusting a valve position or bymodifying a rotation rate of a screw feeder.

The first control system 100 comprises a controller 102 that receivesinput parameter values from an operator through an interface with thefirst control system 100, and also receives sensed parameter values fromsensors coupled to or integral with the physical plant 10. Thecontroller 102 distributes as output parameter values commands to thefirst actuator 18 and the second actuator 20. An input parameter value104 for h₂ provides to the controller 102 the desired height h₂ of thesecond mixture 15 in the second tank 16, an input parameter value 106for dv_(s)/dt provides the desired volumetric flow rate dv_(s)/dt of thesecond mixture 15 out of the second tank 16, and an input parametervalue 108 for ρ_(s) provides the desired density ρ_(s) of the secondmixture 15 out of the second tank 16. A h₁ sensor 110 provides anindication of the height h₁ of the first mixture 13 in the first tank12, a h₂ sensor 112 provides an indication of the height h₂ of thesecond mixture 15 in the second tank 16, a ρ₁₂ sensor 114 provides anindication of the density ρ₁₂ of the first mixture 13, and a ρ_(s)sensor 116 provides an indication of the density ρ_(s) of the secondmixture 15. These indications may be referred to as sensed parametervalues.

In an embodiment, the first control system 100 controls the actuators18, 20 to achieve a desired volumetric flow rate dv₁₂/dt of the firstmixture 13 over the weir 14 into the second tank 16, and a density ρ_(s)of the second mixture 15 out of the second tank 16, independently ofeach other. For example, changing the dv_(s)/dt input parameter value106 causes the controller 102 to control the actuators 18, 20 so thatthe actual volumetric flow rate dv₁₂/dt of the first mixture 13 over theweir 14 into the second tank 16 changes until it substantially equalsthe dv_(s)/dt input parameter value 106. Actual values may also bereferred to as nominal values in some contexts by those skilled in thecontrol systems art. However, the density ρ_(s) of the second mixture 15as it leaves the second tank 16 remains substantially unchanged becausethe density and flow rate parameters are controlled independently.Similarly, changing the ρ_(s) input parameter value 108 causes thecontroller 102 to alter the control signals to actuators 18, 20 untilthe sensed density ρ_(s) of the second mixture 15 as read by the ρ_(s)sensor 116 substantially equals the ρ_(s) input parameter value 108.However, the volumetric flow rate dv₁₂/dt of the first mixture 13 overthe weir 14 into the second tank 16 remains substantially unchangedbecause the flow rate and density parameters are controlledindependently.

Turning now to FIG. 3, a flow modulator 150 for use with one or moreembodiments of the first control system 100 is depicted. The flowmodulator 150 comprises a first flow modulator 152 and a second flowmodulator 154 to modulate the first actuator 18 and the second actuator20, respectively. In some embodiments, the first flow modulator 152 andthe second flow modulator 154 may be combined in an integrated modulatorunit 155 or combined in a single functional block, such as, for example,within a computer programmed to modulate the first actuator 18 and thesecond actuator 20. The first and second flow modulator 152, 154 providea mechanism for modulating or converting between the preferred controlparameters, for example volumetric rate and mass flow rate, and thephysical controls of the physical system, for example control signals tothe first actuator 18 and the second actuator 20. Both the first flowmodulator 152 and the second flow modulator 154 receive from anothercomponent of the controller 102 a commanded first volume flow ratesignal 151 for dv_(in)/dt and a commanded first mass flow rate signal153 for dm_(in)/dt. For example, the commanded first volume flow ratedv_(in)/dt signal 151 and the commanded first mass flow rate dm_(in)/dtsignal 153 may be received from a flow regulator 200 to be discussedhereinafter, or from some other component of the controller 102. Thecommanded first volume flow rate dv_(in)/dt signal 151 corresponds to adesired combined volumetric in-flow rate of materials, for example drycement and carrier fluid, from the first actuator 18 and the secondactuator 20 into the first tank 12. The commanded first mass flow ratedm_(in)/dt signal 153 represents a desired combined mass in-flow rate ofthese materials from the first actuator 18 and the second actuator 20into the first tank 12.

In an embodiment, the flow modulator 150 generates an actuator₁ signalto control the first actuator 18, and this actuator₁ signal may beexpressed mathematically as proportional to a function f₁ as follows:actuator₁ signal∝f ₁ =[dm _(in) /dt−(dv _(in)/dt)ρ_(m2)]/(ρ_(m1)−ρ_(m2))  (1)Similarly, the flow modulator 150 generates an actuator₂ signal tocontrol the second actuator 20, and the actuator₂ signal may beexpressed mathematically as proportional to a function f₂ as follows:actuator₂ signal∝f ₂=[(dv _(in) /dt)ρ_(m1) −dm _(in)/dt]/(ρ_(m1)−ρ_(m2))  (2)where dm_(in)/dt is the combined mass flow rate entering the tank,dv_(in)/dt is the combined volume flow rate entering the tank, ρ_(m1) isthe density of the first material, for example dry cement, flowing intothe first tank 12 from the first actuator 18, and where ρ_(m2) is thedensity of the second material, for example water, flowing into thefirst tank 12 from the second actuator 20. One skilled in the art willreadily be able to determine a suitable first constant ofproportionality for equation (1) to drive the first actuator 18 and asuitable second constant of proportionality for equation (2) to drivethe second actuator 20 in a particular embodiment. In an embodiment, theactuator₁ and actuator₂ signals may be conditioned by one or morecomponents (not shown) between the flow modulator 150 and the actuators18, 20 to conform the actuator₁ and actuator₂ signals to a non-linearresponse of one or more of the actuators 18, 20.

Turning now to FIG. 4, a flow regulator 200 for use with one or moreembodiments of the first control system 100 is depicted. The flowregulator 200 comprises a first flow regulator 202 and a second flowregulator 204. In some embodiments, the first flow regulator 202 and thesecond flow regulator 204 may be combined in an integrated unit 205 orcombined in a single functional block, such as, for example, within acomputer programmed to perform flow regulation. The first flow regulator202 receives from another component of the controller 102 a commandedvolumetric flow rate signal 201 for dv₁₂/dt of the first mixture 13 overthe weir 14 into the second tank 16, and a sensed height h₁ of the firstmixture 13 in the first tank 12 from the h₁ sensor 110. The first flowregulator 202 may receive the commanded volumetric flow rate dv₁₂/dtsignal 201, for example, from a flow₁ state feedback controller withcommand feed forward 250, to be discussed hereinafter, or from someother component of the controller 102. The first flow regulator 202generates the commanded first volumetric flow rate dv_(in)/dt signal 151that is received as a commanded value by the flow modulator 150 or bythe first and second flow modulators 152, 154 as described above withreference to FIG. 3.

The second flow regulator 204 receives from another component of thecontroller 102 a commanded mass flow rate signal 203 for dm₁₂/dt of thefirst mixture 13 over the weir 14 into the second tank 16, a sensedheight h₁ of the first mixture 13 in the first tank 12 from the h₁sensor 110, and a sensed density ρ₁₂ of the first mixture 13 from sensor114. The second flow regulator 202 may receive the commanded mass flowrate dm₁₂/dt signal 203, for example, from a flow₂ state feedbackcontroller with command feed forward 252, to be discussed hereinafter,or from some other component of the controller 102. The second flowregulator 204 generates the commanded first mass flow rate dm_(in)/dtsignal 153 that is received as a commanded value by the flow modulator150 or by the first and second flow modulators 152, 154 described abovewith reference to FIG. 3.

In an embodiment, the function of the first flow regulator 202 may beexpressed mathematically as:commanded dv _(in) /dt=F(h ₁)(1−K _(v))+K _(v)(commanded dv ₁₂ /dt)  (3)and the function of the second flow regulator 204 may be expressedmathematically as:commanded dm _(in) /dt=F(h ₁)ρ₁₂(1−K _(m))+K _(m)(commanded dm ₁₂/dt)  (4)where F(h₁) is a non-linear function of the height h₁ of the firstmixture 13 in the first tank 12 and provides an estimate of a volumetricflow rate dv₁₂/dt of the first mixture 13 over the weir 14 into thesecond tank 16; where commanded dv₁₂/dt is the commanded volumetric flowrate dv₁₂/dt signal 201; where commanded dm₁₂/dt is the commanded massflow rate dm₁₂/dt signal 203; where ρ₁₂ is an indication of density ofthe first mixture 13 in the first tank 12 based on the input from theρ₁₂ sensor 114; and where K_(v) and K_(m) are constants ofproportionality that are greater than zero. The values for K_(v) andK_(m) may be chosen through a closed form solution and/or iteratively soas to minimize overall response time while maintaining stability anddesired flow rate and density trajectories during transition phases. Anexemplary non-linear function F(h₁) for volumetric flow rate over arectangular weir is given in Engineering Fluid Mechanics, 5^(th)Edition, by Roberson and Crowe, published by Houghton Mifflin, 1993, andmay be represented as:dv ₁₂ /dt=F(h ₁)=KL(h ₁ −h _(w))^((3/2))  (5)where L is the length of the weir 14 dividing the tanks 12, 16, K is aflow coefficient that may be empirically determined over a set ofoperating conditions for a specific weir geometry, and h_(w) is aconstant representing the height of the weir.

The volumetric outflow of the first mixture 13 dv₁₂/dt and the massoutflow of the first mixture 13 dm₁₂/dt from the first tank 12 to thesecond tank 16 may be modeled as negative state feedbacks in thephysical plant 10. In equation (3), the effect of the 1×F(h₁) term is tocancel or decouple the negative state feedback associated with dv₁₂/dt,and in equation (4), the effect of the 1×ρ₁₂F(h₁) term is to decouplethe negative state feedback associated with dm₁₂/dt. The control system102 may be more robust as a result of the state feedback decoupling inthe first and second flow regulators 202, 204 because the control system102 only has to correct for errors between desired and actual mass rateand desired and actual volumetric flow rate without having to alsocompensate for the first mixture 13 leaving the first tank 12.

Turning now to FIG. 5A, a flow, state feedback controller with commandfeed forward 250 and a flow₂ state feedback controller with command feedforward 252 for use with one or more embodiments of the first controlsystem 100 are depicted. The flow₁ state feedback controller withcommand feed forward 250 receives the h₂ input parameter value 104 andthe dv_(s)/dt input parameter value 106 from an operator interfacingwith the first control system 100, and also receives an indication ofthe height h₂ of the second mixture 15 in the second tank 16 from the h₂sensor 112. Based on these inputs, the flow₁ state feedback controllerwith command feed forward 250 produces the commanded volumetric flowrate dv₁₂/dt signal 201 that is received as a commanded value by thefirst flow regulator 202 described above with reference to FIG. 4. Theflow₁ state feedback controller with command feed forward 250 may alsobe referred to as a height controller with command feed forward.

The flow₂ state feedback controller with command feed forward 252receives the dv_(s)/dt input parameter value 106 and the ρ_(s) inputparameter value 108, from an operator interfacing with the first controlsystem 100, and also receives an indication of the density ρ_(s) of thesecond mixture 15 in the second tank 16 from the ρ_(s) sensor 116. Basedon these inputs, the flow₂ state feedback controller with command feedforward 252 produces the commanded mass flow rate dm₁₂/dt signal 203that is received as a commanded value by the second flow regulator 204described above with reference to FIG. 4. The flow₂ state feedbackcontroller with command feed forward 252 may also be referred to as adensity controller with command feed forward. The flow state feedbackcontrollers with command feed forward 250, 252 receive different inputsbut serve the same purpose of providing a commanded value signal to aflow regulator 202, 204. In some embodiments, the flow₁ state feedbackcontroller with command feed forward 250 and the flow₂ state feedbackcontroller with command feed forward 252 may be combined in anintegrated unit 251 or combined in a single functional block, such as,for example, within a computer programmed to perform the indicatedfunctions.

Turning now to FIG. 5B, a block diagram shows processing details of oneembodiment of the flow₁ state feedback controller with command feedforward 250. A first summation component 253, represented by the Σsymbol within a circle as is conventional in mathematical notation,determines a first error term e₁(t) by negatively summing the indicationof the height h₂ of the second mixture 15 in the second tank 16 from theh₂ sensor 112 with the h₂ input parameter value 104. The inputs to anysummation component, for example the first summation component 253, arepositively or negatively summed together to determine the output of thesummation component. Specifically, the inputs associated with a “+”(plus) sign are positively summed, while the inputs associated with a“−” (minus) sign are negatively summed. The output of the firstsummation component 253, namely the first error term e₁(t), is thenprocessed by a first proportional-integral (PI) controller 255 having again K_(p1) for a first proportional component 254 and an integral gainK_(i1) for a first integral component 256 associated with a firstintegration factor 258, as represented by 1/S inside the box, as isconventional in control system art to suggest integration. Theproportional and integral operations on the first error term e₁(t) arepositively summed by a second summation component 259. A twenty-fifthsummation component 257 sums the output of the second summationcomponent 259 with the dv_(s)/dt input parameter value 106 of the secondmixture 15 out of the second tank 16. The output of the twenty-fifthsummation component 257 is the commanded volumetric flow rate dv₁₂/dtsignal 201 that is received as a commanded value by the first flowregulator 202 described above with reference to FIG. 4. The temporalresponse of the flow, state feedback controller with command feedforward 250, represented as a function u₁(t), may be expressedmathematically as:commanded dv ₁₂ /dt=u ₁(t)=dv _(s) /dt+K _(p1) e ₁(t)+K _(i1) ∫e₁(t)dt  (6)

The outflow of the second mixture 15 from the second tank 16 may bemodeled as negative state feedback in the physical plant 10. In equation(6), the effect of the dv_(s)/dt term, which is the command feed forwardterm, is to decouple the negative state feedback associated with theoutflow of the second mixture 15 from the second tank 16. While this maynot be strict state feedback decoupling, the same benefits of robustnessmay be at least partially obtained using this technique.

Turning now to FIG. 5C, a block diagram shows processing details of oneembodiment of the flow₂ state feedback controller with command feedforward 252. A third summation component 261 determines a second errorterm e₂(t) by negatively summing the indication of the density ρ_(s) ofthe second mixture 15 in the second tank 16 from the ρ_(s) sensor 116with the ρ_(s) input parameter value 108. The second error term e₂(t) isthen processed by a second PI controller 263 having a gain K_(p2) for asecond proportional component 260 and an integral gain K_(i2) for asecond integral component 262 associated with a second integrationfactor 264. The proportional and integral operations on the second errorterm e₂(t) are positively summed by a fourth summation component 265. Atwenty-sixth summation component 287 sums the output of the fourthsummation component 265 with the output of a first multiplier component266 and the output of a second multiplier component 267. The firstmultiplier component 266 output equals the ρ_(s) input parameter value108 multiplied by (dh₂/dt)A₂, where A₂ is the cross-sectional area ofthe second tank 16 and dh₂/dt is the height rate of change of the secondmixture 15 in the second tank 16. In an embodiment, the first multipliercomponent 266 may be omitted, as for example when the parameter dh₂/dtis not readily available. The second multiplier component 267 outputsthe product of the ρ_(s) input parameter value 108 multiplied by theinput parameter value 106 of dv_(s)/dt of the second mixture 15 out ofthe second tank 16. The output of the twenty-sixth summation component287 is the commanded mass flow rate dm₁₂/dt signal 203 that is receivedas a commanded value by the second flow regulator 204 described abovewith reference to FIG. 4. Thus, the temporal response of the flow₂ statefeedback controller with command feed forward 252, represented as afunction u₂(t), may be expressed mathematically as:commanded dm ₁₂ /dt=u ₂(t)=inputρ_(s){(dh ₂ /dt)A ₂ +dv _(s) /dt}+K_(p2) e ₂(t)+K _(i2) ∫e ₂(t)dt  (7)

The outflow of mass from the second tank 16 may be modeled as negativestate feedback in the physical plant 10. In equation (7), the effect ofthe ρ_(s)(dv_(s)/dt) term, the command feed forward term, is to decouplethe negative state feedback associated with the outflow of mass from thesecond tank 16. While this may not be strict state feedback decoupling,the same benefits of robustness may be at least partially obtained usingthis technique. An additional refinement of this relaxed state feedbackdecoupling technique may be obtained by decoupling the effect of a lossof mass associated with changes of height of the second mixture 15 inthe second tank 16. In equation (7), the ρ_(s)(dh₂/dt)A₂ term alsocontributes to decoupling the negative state feedback associated withthe outflow of mass from the second tank 16. The dh₂/dt factor may bedetermined from a series of indications of the height of the secondmixture 15 in the second tank 16 or by other means such as an estimateof dh₂/dt produced by a height observer as discussed below.

One skilled in the art will recognize that the results of the analysisof the flow₁ state feedback controller with command feed forward 250 andthe flow₂ state feedback controller with command feed forward 252 abovemay be applied to digital signals as well as analog signals. Forexample, analog parameters such as the indication of density ρ_(s) ofthe second mixture 15 in the second tank 16 from the ρ_(s) sensor 116may be converted by an analog-to-digital converter (D/A converter) to adigital signal. Similarly, analog outputs may be produced by adigital-to-analog converter (A/D converter), optionally combined with anamplifier to provide sufficient power to drive an electromechanicaldevice, converting a digital control signal to an analog control signalsuitable for controlling the first actuator 18 and the second actuator20.

Turning now to FIG. 6, the major components of the first control system100 are depicted coupled together and to the actuators 18, 20 of thephysical plant 10 described above with reference to FIG. 1. Inparticular, FIG. 6 depicts an embodiment of the first control system 100of FIG. 2, showing the various components that may comprise thecontroller 102.

Referring to the right side of the drawing, the first and second flowmodulators 152, 154, as described above with reference to FIG. 3, areshown cross-coupled and connected to control the first and secondactuators 18, 20. The first flow modulator 152 receives the commandedfirst volume flow rate dv_(in)/dt signal 151 from the first flowregulator 202 and the commanded first mass flow rate dm_(in)/dt signal153 from the second flow regulator 204 to generate an actuator₁ controlsignal to control the first actuator 18. The second flow modulator 154receives the commanded first volume flow rate dv_(in)/dt 151 signal fromthe first flow regulator 202 and the commanded first mass flow ratedm_(in)/dt 153 signal from the second flow regulator 204 to generate anactuator₂ control signal to control the second actuator 20. Thus, thefirst and second flow modulators 152, 154 may be said to control theactuators 18, 20 based on the dv_(in)/dt and dm_(in)/dt systemparameters, which are not the parameters desired to be controlled.Instead, these system parameters dv_(in)/dt and dm_(in)/dt areassociated closely with the operation of the first and second actuators18, 20 by the flow modulators 152, 154. However, it is desired tocontrol other system parameters, namely the input parameter values,which are derived from the states of the first and second actuators 18,20 and the characteristics of the physical plant 10, for example thecross-sectional area of the first and second tanks 12, 16, and are timelagged with respect to the the states of the actuators 18, 20. Thepossible states of the actuators 18, 20 depend on the actuator type. Forexample, a valve opens or closes, whereas a screw feeder turns atdifferent speeds. The various other components of the controller 102,which will be discussed herein, enable control of the desired systemparameters by bridging between the states of the actuators 18, 20 andthose parameters desired to be controlled.

The first flow regulator 202 receives the commanded volumetric flow ratedv₁₂/dt signal 201 from the flow₁ state feedback controller with commandfeed forward 250 and the h₁ indication from the h₁ sensor 110 togenerate the commanded first volume flow rate dv_(in)/dt signal 151 thatfeeds into the flow modulators 152,154. As discussed above withreference to FIG. 4, the first flow regulator 202 uses the indication ofh₁ to decouple the negative state feedback associated with the flow ofthe first mixture 13 out of the first tank 12. For example, thevolumetric outflow can be determined from F(h₁), as discussed above, andthen the volumetric outflow can be decoupled.

The second flow regulator 204 receives the commanded mass flow ratedm₁₂/dt signal 203 from the flow₂ state feedback controller with commandfeed forward 252, the indication from the h₁ sensor 110, and the ρ₁₂indication from the ρ₁₂ sensor 114 to generate the commanded first massflow rate dm_(in)/dt signal 153 that feeds into the flow modulators 152,154. As discussed above with reference to FIG. 4, the second flowregulator 204 uses the indications of h₁ and of ρ₁₂ to decouple thenegative state feedback associated with the flow of the first mixture 13out of the first tank 12. For example, the mass outflow can becalculated from the product of the indication of ρ₁₂ and the volumetricoutflow dv₁₂/dt, where the volumetric outflow dv₁₂/dt is determined fromF(h₁), and the mass outflow then can be decoupled. The first and secondflow regulators 202, 204 may be said to provide one level of removalfrom the system parameters dv_(in)/dt and dm_(in)/dt directly associatedwith the states of the first and second actuators 18, 20.

The flow₁ state feedback controller with command feed forward 250receives the h₂ input parameter value 104 and the dv_(s)/dt inputparameter value 106 from an operator interfacing with the first controlsystem 100, and also receives the indication of h₂ from the h₂ sensor112 to generate the commanded volumetric flow rate dv₁₂/dt signal 201that feeds into the first flow regulator 202. As discussed above, theflow₁ state feedback controller with command feed forward 250 may bereferred to as a height controller, because it controls the height h₂ ofthe second mixture 15 in the second tank 16. The command feed forwardterm, namely the dv_(s)/dt input parameter value 106, may be consideredto decouple the negative state feedback of volumetric flow out of thesecond tank 16 as discussed above with reference to FIG. 5B.

The flow₂ state feedback controller with command feed forward 252receives the dv_(s)/dt input parameter value 106 and the ρ_(s) inputparameter value 108 from an operator interfacing with the first controlsystem 100, and also receives the indication of ρ_(s) from the ρ_(s)sensor 116 to generate the commanded mass flow rate dm₁₂/dt signal 203that feeds into the second flow regulator 204. As discussed above, theflow₂ state feedback controller with command feed forward 252 may bereferred to as a density controller, because it controls the densityρ_(s) of the second mixture 15 in the second tank 16. The command feedforward term, formed by the product of the ρ_(s) input parameter value108 multiplied by the dv_(s)/dt input parameter value 106, may beconsidered to decouple the negative state feedback of mass flow out ofthe second tank 16 as discussed above with reference to FIG. 5C.

The flow₁ and flow₂ state feedback controller with command feed forward250, 252 may be said to provide yet another level of removal from thesystem parameters dv_(in)/dt and dm_(in)/dt directly associated with thestates of the first and second actuators 18, 20, and provide the desiredresponsiveness to the parameters desired to be controlled.

It will be readily appreciated by one skilled in the art that portionsof the control components described above may be combined, for examplewithin a computer program implementing the functional blocks of thecontrol components.

In operation, as set out in the logic flow diagram of FIG. 7, the firstcontrol system 100 disclosed herein may, for example, be used to controla physical plant 10 comprising a cement slurry mixing system thatprovides cement slurry for cementing a tubular casing into a well bore.During such an operation, the operator may want the first control system100 to provide a cement slurry having a desired density ρ_(s), whilepumping the cement slurry out of the second tank 16 at a desiredvolumetric flow rate dv_(s)/dt, and while maintaining the height of thecement slurry in the second tank 16 at a desired height h₂.

The process begins at block 400 in which the well bore servicingequipment, including the physical plant 10 and the first control system100, is brought to the well bore site and assembled. The discharge line32 of FIG. 1 is coupled to the well bore, for example, using connectedlengths of pipe, one end of which connects to a manifold or header atthe well bore. Vessels containing blended dry cement and water insufficient quantities are positioned to continuously supply the firstfeed line 28 and the second feed line 30 of FIG. 1.

Once the equipment has been set up at the well site, the processproceeds to block 402 where an operator provides input parameter valuesto the controller 102, for example through a console or lap top computercoupled to the controller 102, for example via a serial cable or awireless link. The input parameter values may include the h₂ input 104,the dv_(s)/dt input 106, and the ρ_(s) input 108. In operation, thecontroller 102 will act to control the first and second actuators 18,20, for example valves, such that the corresponding actual quantities ofρ_(s), dv_(s)/dt, and h₂ approach or equal the input parameter values.

The process proceeds to block 404 where the operator engages thecontroller 102. The process continues hereinafter along two independentbut at least partially coupled paths. Proceeding to block 406, thecontroller 102 actively controls the first and second actuators 18, 20in accordance with the input parameter values and the sensed conditionsof the physical plant 10. When the controller 102 is first engaged, itis likely that the first and second tanks 12, 16 will be empty. In thiscase, the controller 102 may open one of the first or second actuators18, 20 fully open and then open the other of the first or the secondactuators 18, 20 so as to achieve the desired cement slurry density asindicated by the ρ_(s) input parameter value 108. The controller 102continually determines and updates the actuator control signals that areoutput to the first and second actuators 18, 20, and the controller 102may be said to be operating within a control loop represented by blocks406 and 408. The controller 102 remains in this control loop 406, 408while compensating for changes of indications returned from the physicalplant 10, for example the sensed density ρ_(s) of the cement slurry inthe second tank 16 according to sensor 116, and for changes of inputvalues, for example the h₂ input parameter value 104.

The process also concurrently proceeds along a second branch to block410 where the physical plant 10 begins to pump cement slurry via theoutflow pump 22, through the discharge line 32, through the coupledpipes, and downhole into the well bore. As the flow rate dv_(s)/dt ofthe cement slurry exiting the second tank 16 via the outflow pump 22changes, for example due to fluctuations in electrical power driving theoutflow pump 22 or due to fluctuations in well bore back pressure, thecontroller 102 adjusts and maintains the actual physical parameters ofthe physical plant 10 to approach or equal the input parameter values.

The process proceeds to block 412 where the operator may modify an inputparameter value, for example the ρ_(s) input 108. This change causes thecontroller 102 to change the control signals that are output to thefirst and second actuators 18, 20 in the control loop 406, 408, forexample by further opening the first actuator 18 and further closing thesecond actuator 20. The coupling between this action of modifying aninput parameter value in block 412 and the control loop in blocks 406,408 is indicated by a dotted line in FIG. 7.

The process proceeds to block 414 where the operator stops thecontroller 102. This action in block 414 will affect the control loop inblocks 406, 408 as indicated by a dashed line in FIG. 7. Specifically,stopping the controller 102 in block 414 causes the control loop inblocks 406, 408 to be exited. The process proceeds to block 416 wherethe physical plant 10 may be disconnected from the supply lines 28, 30for water and dry cement, respectively, and the discharge line 32 may bedisconnected from the pipes coupling the physical plant 10 to the wellbore. The other equipment may be disassembled, the well bore may beclosed in for a period to allow the cement to set, and the equipment ofthe physical plant 10 may be flushed and/or cleaned. The process exitsat block 418.

In an embodiment, the indication of the height h₁ of the first mixture13 in the first tank 12, and the indication of the height h₂ of thesecond mixture 15 in the second tank 16 are provided by two heightobserver components, which estimate rather than directly sense h₁ andh₂. In another embodiment, a single height observer may be employed toprovide an indication of the height h₂ of the second mixture 15 in thesecond tank 16. In yet another embodiment, a single height observer maybe employed to provide an indication of the height h₁ of the firstmixture 13 in the first tank 12, for example in a physical plant 10having only one tank.

Under field conditions, the height indications h₁, h₂ provided by the h₁sensor 110 and the h₂ sensor 112 may be subject to various errors, forexample height oscillations due to movement of the physical plant 10onboard a floating platform or ship. Additionally, the stirring actionof the first stirrer 24 and second stirrer 26 may significantly agitatethe level surface of the first mixture 13 in the first tank 12 and ofthe second mixture 15 in the second tank 16, introducing variations inthe height indications h₁ and h₂. The introduction of the first andsecond materials into the first tank 12, for example dry cement and/orwater, may introduce further variations in the height indication h₁provided by the h₁ sensor 110. All of these surface height variationsmay be analyzed as noise in the height signals. It may be desirable toemploy estimated height indications of h₁ and h₂ rather than propagatethe noise or oscillation that may be present in the indications ofdirect sensors, for example the h₁ sensor 110 and the h₂ sensor 112,into the controller 102.

Generally, a height observer is implemented as a dynamic control systemto obtain an estimated height of the mixture in the tank in real time.First, this estimate of the mixture height is compared to the measuredmixture height to obtain a height error. Then this mixture height erroris used to drive the estimated mixture height to an actual mixtureheight through the use of a proportional-integral type controller, alsoreferred to as a PI controller. By setting the gains of the PIcontroller, the noise and oscillations of the mixture in the tank can besubstantially removed from the mixture height estimate while trackingthe actual value of the mixture height. The height observer according tothe present disclosure reduces the negative effects of noise and poorsensor performance due to environmental effects such as cement dust inthe air or tank oscillations from the height readings. This heightobserver reflects the state of the actual mixture height withsubstantially no time lag.

Turning now to FIG. 8A, a block diagram depicts a general heightobserver 270. The height observer 270 includes a height PI controllercomponent 272 and a height tank model component 274. An estimated heightnegative feedback term 273 is negatively summed by a fifth summationcomponent 269 with a sensed height input 271 to determine a third errorterm e₃(t). As shown in FIG. 8A, the estimated height negative feedbackterm 273 is the same signal as the final output produced by the generalheight observer 270, an estimated height 277, as described herein. Whilethe value of the estimated height negative feedback term 273 and theestimated height 277 are the same, different labels are used herein todistinguish the different functions to which the signals are applied.The estimated height negative feedback term 273 is fed back into theheight observer 270 as an input, thereby creating an estimating controlloop, which yields a more accurate estimated height 277 each timethrough. The third error term e₃(t) is processed by the height PIcontroller component 272 having a gain K_(p3) for a third proportionalcomponent 275 and an integral gain K_(i3) for a third integral component276 associated with a third integration factor 278. The proportional andintegral operations on the third error term e₃(t) are then positivelysummed by a sixth summation component 280. The temporal response of theheight PI controller component 272 to the third error term e₃(t) input,represented by the function u₃(t), may be expressed mathematically as:u ₃(t)=K _(p3) e ₃(t)+K _(i3) ∫e ₃(t)dt  (8)

The output of the height PI controller component 272 may be employed asan estimate of system errors, for example the difference between thedesired and actual volumetric flow rates as well as other parameterestimations, for example the cross sectional area A₁ of the first tank12. These errors may be collected and referred to as a disturbance,where the disturbance accounts for the discrepancies between what isdemanded from the first control system 100 and what the first controlsystem 100 actually produces, unmeasured quantities such as air flowsinto or out of the subject tank, and inaccuracies of estimates of systemparameters. When the physical plant 10 mixes dry cement, there may bedifferences between the desired cement rate and the actual cement rate,because the cement delivery may be inconsistent. Differences between thedesired cement rate and the actual cement rate may be accommodated bythe disturbance. The output of the height PI controller component 272may be referred to as a first disturbance estimate 281. The firstdisturbance estimate 281 may be fed back into the controller 102 toprovide disturbance decoupling. Disturbance decoupling is described inmore detail hereinafter.

The output of the height PI controller component 272 (i.e., the outputfrom the sixth summation component 280) is positively summed by aseventh summation component 282 with one or more height feed forwardinputs 283, for example volumetric flow rates such as the differencebetween the commanded volumetric flow rate dv_(in)/dt into the firsttank 12 and the volumetric flow rate dv₁₂/dt of cement slurry out of thefirst tank 12 over the weir 14 into the second tank 16. The performanceof the height observer 270 is expected to be improved by using heightfeed forward inputs 283 as compared to the performance of moretraditional controllers, which only employ feedback terms with no feedforward inputs. Because the height of a mixture in a tank, for examplethe height h₁ of the first mixture 13 in the first tank 12, willincrease or decrease due to a net positive or negative volumetric flowrate into the tank, the estimate of the height of the mixture in thetank depends upon the net positive or negative volumetric flow rate intothe tank: the height feed forward inputs 283. The height feed forwardinputs 283 may be summed either negatively or positively at the seventhsummation component 282. The output of the seventh summation component282 conforms to a volumetric flow rate that may be represented generallyas dv/dt.

The height tank model component 274 multiplies the volumetric flow rateoutput of the seventh summation component 282 by a fourth integralcomponent 284 associated with a fourth integration factor 286. Thefourth integral component 284 corresponds to the inverse of thecross-sectional area of the tank, represented by “1/A” in the block forthe fourth integral component 284. Multiplying a volumetric flow rateterm (dv/dt) by the inverse of an area term, for example the inverse ofa cross-sectional tank area expressed as 1/A, results in a velocity term(dx/dt), or more particularly in the case of a height controller, anestimated height rate of change dh/dt 285. The intermediate result ofthe estimated height rate of change dh/dt 285 may be used by othercomponents in the controller 102, for example by the flow₂ statefeedback controller with command feed forward 252.

Integrating the velocity term via the fourth integration factor 286results in a displacement x, or in the present case a height h. Thus,the output of the height tank model component 274 is an estimated height277 of the mixture in the tank. One skilled in the art will recognizethat the results of the analysis of the height observer 270 above may beapplied to digital signals as well as analog signals. For example,analog parameters such as the sensed height input 271 may be convertedby an analog-to-digital converter (A/D converter) to a digital signal.Similarly, analog outputs may be produced by a digital-to-analogconverter (D/A converter), optionally combined with an amplifier toprovide sufficient power to drive an electromechanical device,converting a digital control signal to an analog control signal suitablefor controlling the first actuator 18 and the second actuator 20. Heightobservers, such as the height observer 270, are further disclosed inrelated U.S. patent application Ser. No. 11/029,072, entitled “Methodsand Systems for Estimating a Nominal Height or Quantity of a Fluid in aMixing Tank While Reducing Noise,” filed Jan. 4, 2005, which isincorporated herein by reference for all purposes.

Turning now to FIG. 8B, a block diagram shows a portion of oneembodiment of the first control system 100 depicted in FIG. 2 and FIG.6, incorporating a first height observer 270-a and a second heightobserver 270-b rather than directly using the h₁ sensor 110 and the h₂sensor 112. In the embodiment depicted in FIG. 8B, the h₁ sensor 110provides an indication of the height h₁ of the first mixture 13 in thefirst tank 12 to the first height observer 270-a, which is negativelysummed by a ninth summation component 269-a with a first estimatedheight negative feedback term 273-a of the first mixture 13 in the firsttank 12 to obtain an error term e_(3-a)(t) that is fed into a firstheight PI controller component 272-a. The output of the first height PIcontroller component 272-a is positively summed by a tenth summationcomponent 282-a with a height feed forward input 283-a from an eleventhsummation component 279, and the output of the tenth summation component282-a is processed by a first height tank model component 274-a toproduce an estimated height 277-a of the first mixture 13 in the firsttank 12. The first proportional factor 284 (i.e., 1/A) for the firstheight tank model component 274-a employs the cross-sectional area A₁ ofthe first tank 12. The estimated height 277-a of the first mixture 13 inthe first tank 12, which may be referred to as an indication of theheight of the first mixture in the first tank 12, is output to the firstflow regulator 202 and also to the second height observer 270-b. Notethat the estimated height 277-a and the first estimated height negativefeedback term 273-a are the same signals but are identified by differentlabels to point out their different functions in the controller 102. Theoutput from the first height PI controller component 272-a, as a firstdisturbance estimate signal 281-a, optionally may be negatively summedwith the output of the first flow regulator 202 by the eleventhsummation component 279 to provide disturbance decoupling to thedetermination of the commanded volumetric flow rate dv_(in)/dt ofmaterial into the first tank 12. In another embodiment, disturbancedecoupling is provided by a second disturbance estimate signal 281-bthat is negatively summed with the output of the first flow regulator202 by the eleventh summation component 279. The second disturbanceestimate signal 281-b is determined by the second height observer 270-b,as discussed in more detail below. The commanded volumetric flow ratedv_(in)/dt output by the eleventh summation component 279 is thecommanded first volume flow rate dv_(in)/dt signal 151 to the flowmodulator 150 or to the first and second flow modulators 152, 154 asdescribed above with reference to FIG. 3.

The h₂ sensor 112 provides an indication of the height h₂ of the secondmixture 15 in the second tank 16 to the second height observer 270-b,which is negatively summed by a twelfth summation component 269-b with asecond estimated height negative feedback term 273-b of the secondmixture 15 in the second tank 16 to obtain an error term e_(3-b)(t) thatis fed into a second height PI controller component 272-b. The output ofthe second height PI controller component 272-b is positively summed bya thirteenth summation component 282-b with a second height feed forwardinput 283-b output by a first calculation component 268. The firstcalculation component 268 outputs the result of the function F(h₁) basedon the estimated height 277-a of the first mixture 13 in the first tank12. In an embodiment, the value of F(h₁) is calculated by the first flowregulator 202 and provided to the thirteenth summation component 282-bas the second height feed forward input 283-b. In this embodiment, thefirst calculation component 268 is not employed. The output of thethirteenth summation component 282-b is then processed by a secondheight tank model component 274-b to produce a second estimated height277-b of the second mixture 15 in the second tank 16. The firstproportional factor 284 (i.e., 1/A) for the second height tank modelcomponent 274-b employs the cross-sectional area A₂ of the second tank16. Note that the second estimated height 277-b and the second estimatedheight negative feedback term 273-b are the same signals but areidentified by different labels to point out their different functions inthe controller 102.

Within the second height observer 270-b, the output from the secondheight PI controller component 272-b provides the second disturbanceestimate signal 281-b that is negatively summed by the eleventhsummation component 279, as previously discussed. In an embodiment, thesecond disturbance estimate signal 281-b may provide a more accurateestimate of the volumetric rate disturbance in the system 100 ascompared to the first disturbance estimate signal 281-a because theheight of the second mixture 15 in the second tank 16 varies more thanthe height of the first mixture 13 in the first tank 12. Additionally,in the two tank system of FIG. 1, the surface of the first mixture 13 inthe first tank 12 may not be level but may slope downwardly from thepoint where the actuators 18, 20 dispense the materials to the weir 14.The second estimated height 277-b of the second mixture 15 in the secondtank 16, which may be referred to as an indication of the height h₂ ofthe second mixture 15 in the second tank 16, is fed into the flow, statefeedback controller with command feed forward 250 as an input, and thecontrol process is repeated. The estimated height rate of change dh/dt277 of the second height observer 270-b, dh₂/dt, is used for statefeedback decoupling by the flow₂ state feedback controller with commandfeed forward 252.

The advantages of the height observer 270, such as, for example,attenuation of noise and improvement of poor sensor performance,estimation of a disturbance term, and estimation of a parameter rate ofchange, may be obtained in a density observer having a structure relatedto the height observer 270. In an embodiment, the indication of thedensity ρ₁₂ of the first mixture 13 in the first tank 12 and theindication of the density ρ_(s) of the second mixture 15 in the secondtank 16 are provided by two density observer components which estimaterather than directly sense the densities ρ₁₂, ρ_(s) of the mixtures 13,15, respectively. In another embodiment, the density observer may beused to estimate the density of other mixtures or materials in systemsother than the physical plant 10 depicted in FIG. 1.

Turning now to FIG. 9A, a general density observer 290 operable todetermine an estimated density of a mixture in a tank is depicted. Thedensity observer 290 includes a density PI controller component 292 anda density tank model component 294. An eighth summation component 299negatively sums an estimated density negative feedback term 295 with asensed density input 291 to determine a fourth error term e₄(t). Thefourth error term e₄(t) is processed by the density PI controllercomponent 292, which has a gain K_(p4) for a fourth proportionalcomponent 296 and an integral gain K_(i4) for a fifth integral component298 associated with a fifth integration factor 300. In an embodiment,the fourth error term e₄(t) may be processed by a filter instead of bythe density PI controller component 292, such as an analog filter or adigital filter, to introduce desirable time lags or to attenuate and/oramplify certain frequency components of the fourth error term e₄(t).

The output of the density PI controller component 292 may be employed asan estimate of system errors, for example the difference between thedesired and actual mass flow rates as well as other parameterestimations. These errors may be collected and referred to as adisturbance, where the disturbance accounts for the discrepanciesbetween what is demanded from the first control system 100 and what thefirst control system 100 actually produces, unmeasured quantities suchas air flows into or out of the subject tank, and inaccuracies ofestimates of system parameters. The output of the density PI controllercomponent 292 may be referred to as a second disturbance estimate 301.The second disturbance estimate 301 may be fed back into the controller102 to provide disturbance decoupling. Disturbance decoupling isdescribed in more detail hereinafter.

The output of the density PI controller component 292, which conforms toa mass flow rate, is summed with one or more density feed forward inputs293, for example a mass flow rate, such as the difference between thecommanded mass flow rate into the first tank 12 and the mass flow rateof cement slurry out of the first tank 12 over the weir 14 into thesecond tank 16, by a third summation component 302. The output of atwelfth multiplication component 435 and the output of a thirteenthmultiplication component 440 are also negatively summed by the thirdsummation component 302. Generally, the density feed forward inputs 293are associated with mass flow into the associated tank and the outputsof the twelfth and thirteenth multiplication components 435, 440 areassociated with mass flow out of the associated tank. The output of thethird summation component 302 is processed by the density tank modelcomponent 294. The density tank model component 294 multiplies theoutput of the third summation component 302 by a sixth integralcomponent 304 associated with a sixth integration factor 306. The sixthintegral component 304 is inversely proportional to the height of themixture times the cross-sectional area of the tank, as represented by“1/hA” in the block for the sixth integral component 304. Note thatdividing a mass flow rate by hA is substantially equivalent to dividingthrough by the volume of the tank resulting in an estimated density rateof change 307 dρ/dt. The height may be provided by a height sensor, forexample the h₁ sensor 110, or the height observer 270. Alternatively,the height may be a fixed constant determined by experimentation toprovide a preferred response rate of the general density observer 290.Integrating this quotient with respect to time results in a density. Theoutput of the density tank model component 294 is thus the estimateddensity 297 of the mixture in the tank. The estimated density negativefeedback term 295 is the same signal as the estimated density 297, butthe estimated density feedback term 295 is fed back into the eighthsummation component 299 at the input to the density observer 290 to beprocessed through the PI controller 292 and tank model component 294 toyield a more accurate estimated density 297 each time through. Theestimated density negative feedback term 295 is multiplied by a factorA(dh/dt) by the twelfth multiplication component 435 to produce aρA(dh/dt) term that is negatively fed back to the third summationcomponent 302 as described above. The ρA(dh/dt) term corresponds to amass rate of change based on changes in the height of the mixture in thetank. The dh/dt factor may be determined from a height sensor or aheight observer. Alternatively, in an embodiment of the system 100 thatprovides no indication or estimate of height of the mixture, the twelfthmultiplication 435 may be absent from the general density observer 290.The estimated density negative feedback term 295 is also multiplied by afactor dv/dt by the thirteenth multiplier component 440 to produce aρ(dv/dt) term that is negatively fed back to the third summationcomponent 302 as described above. The ρ(dv/dt) term corresponds to amass rate of change due to flow of the mixture out of the tank. In anembodiment, the sensed density input 291 may be provided by a densitysensor, for example the ρ₁₂ sensor 114, installed in a recirculationline or an outflow line, such as the discharge line 32, associated withthe tank. Thus, the density of the mixture as measured in therecirculation line may lag several seconds behind the density of themixture in the tank. In this embodiment, a time delay of severalseconds, such as three seconds, for example, may be introduced in theestimated density negative feedback term 295 before negatively summingwith the sensed density 291 at the eighth summation component 299 todetermine the fourth error term e₄(t). This allows the sensed andestimated density to be in the same time reference frame beforedetermining the fourth error term e₄(t). The appropriate time delay maybe readily determined from experimentation by one skilled in the art anddepends upon the viscosity of the mixture and the speed of mixing.

Because the structures of the height observer 270 and the densityobserver 290 are related, one skilled in the art need only determine thegains appropriate to the height observer 270, determine the gainsappropriate to the density observer 290, and configure the two observerstructures 270, 290 accordingly. One skilled in the art will recognizethat the results of the analysis of the density observer 290 above maybe applied to digital signals as well as analog signals. For example,analog parameters such as the indication of density ρ_(s) of the secondmixture 15 in the second tank 16 from the ρ_(s) sensor 116 may beconverted by an analog-to-digital converter (A/D converter) to a digitalsignal. Similarly, analog outputs may be produced by a digital-to-analogconverter (D/A converter), optionally combined with an amplifier toprovide sufficient power to drive an electromechanical device,converting a digital control signal to an analog control signal suitablefor controlling the first actuator 18 and the second actuator 20.

Turning now to FIG. 9B, a block diagram shows a portion of oneembodiment of the first control system 100 incorporating a first densityobserver 290-a and a second density observer 290-b. The ρ₁₂ sensor 114provides an indication of the density ρ₁₂ of the first mixture 13 in thefirst tank 12 to the first density observer 290-a, which is negativelysummed by a fourteenth summation component 299-a with a first estimateddensity negative feedback term 295-a of the first mixture 13 in thefirst tank 12 to obtain an error term e_(4-a)(t). As described above, inan embodiment where the ρ₁₂ sensor 114 is located in a recirculationline, the estimated density negative feedback term 295-a may be delayedseveral seconds before it is input to the fourteenth summation component299-a. The e_(4-a)(t) error term is fed into a first density PIcontroller component 292-a. The output of the first density PIcontroller component 292-a is positively summed with a first densityfeed forward input 293-a from a seventeenth summation component 305 by afifteenth summation component 302-a. The output of a fourteenthmultiplication component 435-a and a fifteenth multiplication component440-a, corresponding to a mass rate of change due to changes in heightin the first tank 12 and the mass flow rate out of the first tank 12,respectively, are negatively summed by the fifteenth summation component302-a The sum output by the fifteenth summation component 302-a isprocessed by a first density tank model component 294-a. Note that thesixth integral component 304 for the first density tank model component294-a employs the cross sectional area A₁ of the first tank 12 and theheight h₁ of the first mixture 13 in the first tank 12. Note that thefirst estimated density negative feedback term 295-a and a firstestimated density 297-a are the same signals but are identified bydifferent labels to point out their different functions in thecontroller 102.

The output from the first density PI controller component 292-a, as thesecond disturbance estimate 301, is negatively summed by the seventeenthsummation component 305 with the output from the second flow regulator204 plus the output generated by a fifth multiplier component 321 todetermine the commanded mass flow rate dm₁₂/dt signal 203. The seconddisturbance estimate 301 provides disturbance decoupling to thedetermination of the commanded mass flow rate dm₁₂/dt signal 203, whilethe fifth multiplier component 321 provides state feedback decoupling.The fifth multiplier component 321 outputs the product formed bymultiplying the first estimated density 297-a of the first mixture 13 inthe first tank 12 by the height rate of change dh₁/dt of the firstmixture 13 in the first tank 12 and by the area A₁ of the first tank 12,obtaining the height rate of change dh₁₁/dt, for example, from theestimated height rate of change 285 output of the first height observer270 a. In another embodiment, the second disturbance estimate 301 may beprovided by the second density observer 290-b.

To estimate the density ρ_(s) of the second mixture 15, the ρ_(s) sensor116 provides an indication of the density ρ_(s) of the second mixture 15in the second tank 16 to the second density observer 290-b. Thisindication of ρ_(s) is summed by an eighteenth summation component 299-bwith a second estimated density negative feedback 295-b of the secondmixture 15 in the second tank 16 to obtain an error term e_(4-b)(t). Asdescribed above, in an embodiment where the ρ_(s) sensor 116 is locatedin a recirculation line or in an outflow line, such as the dischargeline 32, the estimated density negative feedback term 295-b may bedelayed several seconds before it is input to the eighteenth summationcomponent 299-b. The error term e_(4-b)(t) is fed into a second densityPI controller component 292-b. A second density feed forward input293-b, output by a sixth multiplier component 322, is positively summedby a nineteenth summation component 302-b with the output of the seconddensity PI controller component 292-b. The sixth multiplier component322 outputs the product formed by multiplying the first estimateddensity 297-a of the first mixture 13 in the first tank 12 by thevolumetric flow rate dv₁₂/dt of the first mixture 13 into the secondtank 16. The output of a sixteenth multiplication component 435-b and aseventeenth multiplication component 440-b, corresponding to a mass rateof change due to changes in height in the second tank 16 and mass flowrate out of the second tank 16, respectively, are negatively summed bythe nineteenth summation component 302-b. The output of the nineteenthsummation component 302-b is processed by a second density tank modelcomponent 294-b. Note that the sixth integral component 304 for thesecond density tank model component 294-b employs the cross sectionalarea A₂ of the second tank 16 and the height h₂ of the second mixture 15in the second tank 16. The second estimated density 297-b of the secondmixture 15 in the second tank 16 is provided as an input to the flow₂state feedback controller with command feed forward 252 for use ingenerating the commanded mass flow rate dm₁₂/dt, which is the commandedmass flow rate dm₁₂/dt signal 203 provided as a commanded parametervalue to the second flow regulator 204 as described above with referenceto FIG. 4. Note that the second estimated density negative feedback term295-b and the second estimated density 297-b are the same signals butare identified by different labels to point out their differentfunctions in the controller 102.

One skilled in the art will readily appreciate that the components ofthe first control system 100 disclosed above are susceptible to manyalternate embodiments. While several alternate embodiments are disclosedhereinafter, other embodiments are contemplated by the presentdisclosure.

Turning now to FIG. 10A, a portion of a second control system 364 isdepicted. The second control system 364 controls a physical plant 10having a single tank, such as the first tank 12 of FIG. 1, and first andsecond actuators 18, 20. The second control system 364 uses a heightobserver 270, which is substantially similar to that described abovewith respect to FIG. 8A, to estimate the height of the first mixture 13in the first tank 12 and to determine the first disturbance estimate281.

Several components are combined to provide a first height controller 366that is substantially similar to the flow₁ state feedback controllerwith command feed forward 250 described above with reference to FIG. 5B.Specifically, an eighteenth summation component 346 determines a fiftherror term e₅(t) by negatively summing the estimated height 277 producedby the height observer 270 with an h₁ input 340, provided for example byan operator. The fifth error term e₅(t) is processed by a proportionalgain term N_(p) in a seventh multiplication component 344. A thirtiethsummation component 426 sums the output of the seventh multiplicationcomponent 344 positively with the dv_(s)/dt input 106 that providescommand feed forward. A twenty-eighth summation component 422 negativelysums the first disturbance estimate 281 with the output of the thirtiethsummation component 426. The twenty-eighth summation component 422provides the same function as the flow, regulator 202 as described abovewith reference to FIG. 8B. The output of the twenty-eighth summationcomponent 422 is the commanded first volume flow rate dv_(in)/dt signal151 that is fed to the first and second flow modulators 152, 154 andinto the seventh summation component 282 of the height observer 270 asthe volume feed forward inputs 283. A first dv_(s)/dt term 445 is fednegatively into the seventh summation component 282, and the firstdv_(s)/dt term 445 may be obtained either from a flow sensor or may bebased on the dv_(s)/dt input 106. The first flow modulator 152 controlsthe first actuator 18 based on the commanded first volume flow ratedv_(in)/dt signal 151 and the commanded first mass flow rate dm_(in)/dtsignal 153, which is generated as described below with reference to FIG.10B.

Turning now to FIG. 10B, another portion of the second control system364 is depicted. Several components are combined to provide a firstdensity controller 368 that is substantially similar to the flow₂ statefeedback controller with command feed forward 252 described above withreference to FIG. 5C, with the exception that the (dh₂/dt)A₂ termassociated with the first multiplication component 266 is omitted,because there is no second tank 16 in this embodiment. A twentiethsummation component 350 determines a sixth error term e₆(t) bynegatively summing the density estimate 297 produced by the modifieddensity observer 356 with the input ρ_(s) 108. The sixth error terme₆(t) is processed by a proportional gain term K_(m) in a ninthmultiplication component 352. A tenth multiplication component 360multiplies the density estimate 297 by the dv_(s)/dt input 106. Thesecond disturbance estimate 362 is negatively summed with the outputs ofthe ninth and tenth multiplication components 352, 360 by a twenty-ninthsummation component 424. The output of the tenth multiplicationcomponent 360 may be considered to provide command feed forward to adensity controller 368 that comprises the twentieth summation component350, the ninth multiplication component 352, the tenth multiplicationcomponent 360, and the twenty-ninth summation component 424. The outputof the twenty-ninth summation component 424 is the commanded first massflow rate dm_(in)/dt signal 153 that is fed into the first and secondflow modulators 152, 154 and into the density observer 290 as thedensity feed forward inputs 293. The second flow modulator 154 controlsthe second actuator 20 based on the commanded first volume flow ratedv_(in)/dt signal 151 and the commanded first mass flow rate dm_(in)/dtsignal 153, which is generated as described above with reference to FIG.10A.

One of ordinary skill in the art will readily appreciate that nointegral processing is employed in the first height controller 366described above with reference to FIG. 10A or in the density controller368 because the effect of command feed forward and disturbancedecoupling is to reduce steady-state error to substantially zero,thereby transforming the second control system 364 from a first ordernon-linear system (as in the first control system 100), to a first orderlinear system that may be controlled by proportional controllers. Ingeneral, if the disturbances are decoupled or removed using theobservers, for example by the height observer 270 and/or the densityobserver 290, and the command feed forward terms, then a proportional(P) controller may be used with substantially zero steady state error,since the second control system 364 will behave in accordance with alinear first order differential equation. If the disturbances are notdecoupled, it may be preferrable to use a PI controller, so that theintegral term acts to ensure substantially zero steady state error.

Turning now to FIG. 11, a third control system 376 is depicted. Thethird control system 376 controls a physical plant 10 having twoactuators 18, 20; two tanks 12, 16; and only a single sensor, the ρ₁₂sensor 114, measuring the density ρ₁₂ of the first mixture 13 in thefirst tank 12. In this embodiment, the commanded first volume flow ratedv_(in)/dt signal 151 to the first and second flow modulators 152, 154is provided directly by the dv_(s)/dt input 106 without employing a flowregulator or other components as in the first control system 100. Also,this embodiment contains no height controller, so an operator may adjustthe actual volumetric flow rate dv_(s)/dt out of the second tank 16 bycontrolling the outflow pump 22, and thereby maintain the desired heighth₂ of the second mixture 15 in the second tank 16.

The density observer 290 determines the density estimate 297 of thedensity ρ₁₂ of the first mixture 13 in the first tank 12 and the seconddisturbance estimate 362 based on the output of the ρ₁₂ sensor 114 anddensity feed forward inputs 293. Because this embodiment provides noindication of height, the A(dh/dt) term in FIG. 9A that is associatedwith the twelfth multiplier component 435 is omitted. As described abovewith reference to FIG. 9A, in the present embodiment the heightcomponent of the sixteenth integral component 304 is a constant valuechosen to provide a desirable system response time. A twenty-secondsummation component 380 negatively sums the density estimate 297 with aρ₁₂ input parameter value 370 to determine a seventh error term e₇(t),which is multiplied by a proportional gain K_(m) by an eleventhmultiplication component 382. The ρ₁₂ input 370, for example, may beprovided by an operator through an interface. The twenty-secondsummation component 380 and the eleventh multiplication component 382comprise a second density controller 372 that controls the density ρ₁₂of the first mixture 13 in the first tank 12. In steady stateconditions, the density of the second mixture 15 in the second tank 16will follow the controlled density ρ₁₂ of the first mixture 13 in thefirst tank 12. When the ρ₁₂ input 370 is changed, the density ρ_(s) ofthe second mixture 15 in the second tank 16 will substantially equal thedensity ρ₁₂ of the first mixture 13 in the first tank 12 after a timelag.

A twelfth multiplication component 384 multiplies the density estimate297 by the approximate actual volumetric flow dv₁₂/dt. Because h₁ is notmeasured, and the function F(h₁) may not be used, the dv_(s)/dt input106 is used to approximate the actual volumetric flow dv₁₂/dt over theweir 14. The output of the twelfth multiplication component 384 providesstate feedback decoupling to the control system 376 to compensate formass leaving the first tank 12.

The second disturbance estimate 362 is negatively summed with the outputof the eleventh and twelfth multiplication components 382, 384 by atwenty-first summation component 374 to determine the commanded firstmass flow rate dm_(in)/dt signal 153. The commanded first mass flow ratedm_(in)/dt signal 153 is provided to the first and second flowmodulators 152, 154 to control the first and second actuators 18, 20 asdescribed above. The output of the twenty-first summation component 374also provides the density feed forward inputs 293 to the densityobserver 290.

Turning now to FIG. 12, a fourth control system 378 is depicted. Thefourth control system 378 is substantially similar to the third controlsystem 376, with the exception that the fourth control system 378includes the height observer 270 and a second height controller 428. Theheight observer 270 determines the estimated height 277 of the secondmixture 15 in the second tank 16 based on input from the h₂ sensor 112and the height feed forward inputs 283. A second dv_(s)/dt factor 450 isnegatively summed with the height feed forward inputs 283 and the outputof the height PI controller 272 by the seventh summation component 282.The estimated height 277 is negatively summed with the h₂ input 104 by atwenty-third summation component 386 to generate an eighth error terme₈(t). The eighth error term e8(t) is processed by a third PI controller390 to produce a commanded height signal 430, which is summed with thedv_(s)/dt input 106 by a twenty-fourth summation component 388 toproduce the dv_(in)/dt signal 151 provided to the flow modulators 152,154. Based on the dv_(in)/dt signal 151 and the dm_(in)/dt signal 153,the first and second flow modulators 152, 154 control the states of thefirst and second actuators 18, 20 as described above.

While the physical plant 10 controlled by the several embodiments of thecontrol systems 100, 364, 376, and 378 described above included eitherone or two tanks, in other embodiments the control systems may controlmixing parameters of more than two tanks connected in series. In otherembodiments, one or more sensors may be employed selected from the ρ₁₂sensor 114, the ρ_(s) sensor 116, the h₁ sensor 110, the h₂ sensor 112,and a flow rate sensor. Embodiments may use a mixture of sensors with noheight observer 270 or density observer 290. Other embodiments may useone or more sensors with one or more height observers 270 and/or densityobservers 290. While the height observer 270 and the density observer290 are discussed above, the observer concept may be extended to othersensors to reduce noise output from these and other sensors and toobtain the benefits associated with decoupling or removing disturbancesand estimating intermediate rate terms. One skilled in the art willreadily appreciate that coupling among control components may be alteredto simplify processing, as for example combining two summationcomponents in series by directing all the inputs to the two summationcomponents to a single summation component or by replacing twocomponents that calculate the same value by one component and routingthe output of the one component to two destinations. Alternately, in anembodiment, a summation component summing more than two inputs may beexpanded into a series of two or more summation components thatcollectively sum the several inputs.

The controller 102 used in the various control systems 100, 364, 376,and 378 described above may be implemented on any general-purposecomputer with sufficient processing power, memory resources, and networkthroughput capability to handle the necessary workload placed upon it.FIG. 13 illustrates a typical, general-purpose computer system 580suitable for implementing one or more embodiments of the several controlsystem embodiments disclosed herein. The computer system 580 includes aprocessor 582 (which may be referred to as a central processor unit orCPU) in communication with memory devices including secondary storage584, read only memory (ROM) 586, random access memory (RAM) 588,input/output (I/O) 590 devices, and network connectivity devices 592.The processor may be implemented as one or more CPU chips. The commandsor signals output to the first actuator 18 and the second actuator 20may be converted from a digital to an analog signal by a digital toanalog converter (DAC), not shown, or otherwise conditioned to make theactuator signal conformable to a control input to the first actuator 18and a control input to the second actuator 20.

The secondary storage 584 typically comprises one or more disk drives ortape drives and is used for non-volatile storage of data and as anover-flow data storage device if RAM 588 is not large enough to hold allworking data. Secondary storage 584 may be used to store programs thatare loaded into RAM 588 when such programs are selected for execution.The ROM 586 is used to store instructions and perhaps data that are readduring program execution. ROM 586 is a non-volatile memory device whichtypically has a small memory capacity relative to the larger memorycapacity of secondary storage 584. The RAM 588 is used to store volatiledata and perhaps to store instructions. Access to both ROM 586 and RAM588 is typically faster than to secondary storage 584.

I/O devices 590 may include printers, video monitors, liquid crystaldisplays (LCDs), touch screen displays, keyboards, keypads, switches,dials, mice, track balls, voice recognizers, card readers, paper tapereaders, or other well-known input devices. The network connectivitydevices 592 may take the form of modems, modem banks, ethernet cards,universal serial bus (USB) interface cards, serial interfaces, tokenring cards, fiber distributed data interface (FDDI) cards, wirelesslocal area network (WLAN) cards, radio transceiver cards such as GlobalSystem for Mobile Communications (GSM) radio transceiver cards, andother well-known network devices. These network connectivity devices 592may enable the CPU 582 to communicate with an Internet or one or moreintranets. With such a network connection, it is contemplated that theCPU 582 might receive information from the network, or might outputinformation to the network in the course of performing theabove-described method steps. Such information, which is oftenrepresented as a sequence of instructions to be executed using processor582, may be received from and outputted to the network, for example, inthe form of a computer data signal embodied in a carrier wave.

Such information, which may include data or instructions to be executedusing processor 582 for example, may be received from and outputted tothe network, for example, in the form of a computer data baseband signalor signal embodied in a carrier wave. The baseband signal or signalembodied in the carrier wave generated by the network connectivitydevices 592 may propagate in or on the surface of electrical conductors,in coaxial cables, in waveguides, in optical media, for example opticalfiber, or in the air or free space. The information contained in thebaseband signal or signal embedded in the carrier wave may be orderedaccording to different sequences, as may be desirable for eitherprocessing or generating the information or transmitting or receivingthe information. The baseband signal or signal embedded in the carrierwave, or other types of signals currently used or hereafter developed,referred to herein as the transmission medium, may be generatedaccording to several methods well known to one skilled in the art.

The processor 582 executes instructions, codes, computer programs,scripts which it accesses from hard disk, floppy disk, optical disk(these various disk based systems may all be considered secondarystorage 584), ROM 586, RAM 588, or the network connectivity devices 592.

While several embodiments have been provided in the present disclosure,it should be understood that the disclosed systems and methods may beembodied in many other specific forms without departing from the spiritor scope of the present disclosure. The present examples are to beconsidered as illustrative and not restrictive, and the intention is notto be limited to the details given herein, but may be modified withinthe scope of the appended claims along with their full scope ofequivalents. For example, the various elements or components may becombined or integrated in another system or certain features may beomitted, or not implemented.

Also, techniques, systems, subsystems and methods described andillustrated in the various embodiments as discrete or separate may becombined or integrated with other systems, modules, techniques, ormethods without departing from the scope of the present disclosure.Other items shown or discussed as directly coupled or communicating witheach other may be coupled through some interface or device, such thatthe items may no longer be considered directly coupled to each other butmay still be indirectly coupled and in communication, whetherelectrically, mechanically, or otherwise with one another. Otherexamples of changes, substitutions, and alterations are ascertainable byone skilled in the art and could be made without departing from thespirit and scope disclosed herein.

1. A control system for mixing at least two materials in a physicalsystem having two or more tanks, comprising: at least two actuators,each actuator being operable to introduce a material into a first tankto form a first mixture, the first mixture flowing into a second tank toform a second mixture; and a controller operable, based on a commandedinput, to control the at least two actuators to obtain a density ofeither the first mixture or the second mixture and a volume flow rate ofthe second mixture out of the second tank; wherein the density iscontrolled independently from the volume flow rate.
 2. The controlsystem of claim 1, wherein the actuators are selected from the groupconsisting of valves, screw feeders, augurs, and elevators.
 3. Thecontrol system of claim 1, wherein the materials are selected from thegroup consisting of water, carrier fluid, dry cement material, sand, andproppants.
 4. The control system of claim 1 wherein the commanded inputcomprises: a commanded combined volume flow rate of the materialspassing through the actuators into the first tank [commandeddv_(in)/dt]; and a commanded combined mass flow rate of the materialspassing through the actuators into the first tank [commandeddm_(in)/dt].
 5. The control system of claim 4, wherein a first actuatoris controlled by a first signal based on a function f₁ where:f ₁=[commanded dm _(in) /dt−(commanded dv _(in)/dt)×ρ_(m2)]/(ρ_(m1)−ρ_(m2)) wherein: ρ_(m1)=density of a firstmaterial; and ρ_(m2)=density of a second material.
 6. The control systemof claim 4, wherein a second actuator is controlled by a second signalbased on a function f₂ where:f ₂=[(commanded dv _(in) /dt)×ρ_(m1)−commanded dm _(in)/dt]/(ρ_(m1)−ρ_(m2)) wherein: ρ_(m1)=density of a first material; andρ_(m2)=density of a second material.
 7. The control system of claim 4,wherein the commanded dv_(in)/dt is based, at least in part, on avolumetric flow rate command feed forward input [dv_(s)/dt input] andthe commanded dm_(in)/dt is based, at least in part, on a firstindication of density of the first mixture and a first commanded densityof the first mixture or the second mixture.
 8. The control system ofclaim 7, wherein the commanded dv_(in)/dt is further based, at least inpart, on an indication of height of the second mixture in the secondtank and an operator input height of the second mixture.
 9. The controlsystem of claim 7, wherein the commanded dv_(in)/dt is further based, atleast in part, on the quantity:F(h)(1−K _(v))+K _(v)(commanded dv ₁₂ /dt) wherein: K_(v)=a firstconstant of proportionality; F(h)=a non-linear function of a height ofthe first mixture in the first tank; and commanded dv₁₂/dt=a commandedvolume flow rate of the first mixture flowing from the first tank intothe second tank.
 10. The control system of claim 10, wherein thecommanded dv₁₂/dt is based on: a first error term determined bysubtracting an indication of a height of the second mixture in thesecond tank from an operator input desired height of the second mixturein the second tank [h₂ input]; and an operator input desired volume flowrate of the second mixture out of the second tank [dv_(s)/dt input]. 11.The control system of claim 10, wherein the first error term isprocessed by either a first proportional integral controller componentor a first proportional controller component.
 12. The control system ofclaim 7, wherein the commanded dv_(in)/dt is further based, at least inpart, on a volumetric disturbance estimate.
 13. The control system ofclaim 7, wherein the commanded dm_(in)/dt is further based, at least inpart, on the quantity:F(h)(ρ₁₂)(1−K _(m))+K _(m)(commanded dm ₁₂ /dt); wherein: K_(m)=a secondconstant of proportionality; F(h)=a non-linear function of a height ofthe first mixture in the first tank; commanded dm₁₂/dt=a commanded massflow rate of the first mixture flowing from the first tank into thesecond tank; and ρ₁₂=an indication of the density of the first mixturein the first tank.
 14. The control system of claim 13, wherein thecommanded dm₁₂/dt is based on a second error term determined bysubtracting the first indication of density from the first commandeddensity.
 15. The control system of claim 13, wherein the commandeddm₁₂/dt is based on a second error term determined by subtracting asecond indication of density of the second mixture from the firstcommanded density of the second mixture.
 16. The control system of claim15, wherein the second error term is processed by either a secondproportional integral controller component or a second proportionalcontroller component.
 17. The control system of claim 15, wherein thecommanded dm₁₂/dt is further based on a parameter selected from thegroup consisting of: an operator input desired volume flow rate of thesecond mixture out of the second tank [dv_(s)/dt input]; and anindication of a height rate of change of the second mixture in thesecond tank.
 18. The control system of claim 7, wherein the commandeddm_(in)/dt is further based, at least in part, on a mass disturbanceestimate.
 19. The control system of claim 7, wherein the commandeddv_(in)/dt is further based, at least in part, on an indication ofheight of the first mixture in the first tank and a commanded height ofthe first mixture in the first tank.
 20. The control system of claim 19,further comprising: a first height sensor operable to produce a firstsensed height output of the first mixture in the first tank; and a firstheight observer operable to provide the indication of height of thefirst mixture in the first tank based on the first sensed height output,the commanded height, and a commanded volumetric flow rate.
 21. Thecontrol system of claim 20, wherein the first height observer is furtheroperable to provide a first disturbance estimate.
 22. The control systemof claim 20, further comprising: a second height sensor operable toproduce a second sensed height output of the second mixture in thesecond tank; and a second height observer operable to provide anindication of height of the second mixture in the second tank based onthe second sensed height output and on the indication of height of thefirst mixture in the first tank.
 23. The control system of claim 22,wherein the second height observer is further operable to provide thefirst disturbance estimate.
 24. The control system of claim 7, furthercomprising: a first density sensor operable to produce a first senseddensity output of the first mixture in the first tank; and a firstdensity observer operable to provide the indication of density of thefirst mixture in the first tank based on the first sensed densityoutput, the first commanded density of the first mixture, and a massflow rate.
 25. The control system of claim 24, wherein the first densityobserver comprises: a third proportional integral controller responsiveto a third error term determined as the sensed density output minus thefirst indication of density of the first mixture in the first tank; anda tank model having an integrating component and a gain component, thegain component inversely proportional to the area of the first tank andto the height of the first mixture in the first tank, the tank modelresponsive to the output of the third proportional integral controller.26. The control system of claim 25, wherein the first indication ofdensity is delayed before determining the third error term.
 27. Thecontrol system of claim 24, wherein the first density observer isfurther operable to provide a second disturbance estimate.
 28. Thecontrol system of claim 24, further comprising: a second density sensoroperable to produce a second sensed density output of the second mixturein the second tank; and a second density observer operable to provide anindication of the density of the second mixture in the second tank basedon the second sensed density output and the indication of density of thefirst mixture in the first tank.
 29. The control system of claim 28,wherein the second density observer is further operable to provide thesecond disturbance estimate.
 30. A control system for mixing at leasttwo materials in a tank, comprising: at least two actuators, eachactuator being operable to introduce a material into the tank to form amixture; and a controller operable, based on a commanded input, tocontrol the at least two actuators to obtain a density of the mixtureand a volume flow rate of the mixture out of the tank; wherein thedensity is controlled independently from the volume flow rate.
 31. Thecontrol system of claim 30, wherein the commanded input comprises: acommanded combined volume flow rate of the materials passing through theactuators into the tank [commanded dv_(in)/dt]; and a commanded combinedmass flow rate of the materials passing through the actuators into thetank [commanded dm_(in)/dt].
 32. The control system of claim 31, whereina first actuator is controlled by a first signal based on a function f₁,where:f ₁=[commanded dm _(in) /dt−(commanded dv _(in)/dt)×ρ_(m2)]/(ρ_(m1)−ρ_(m2)) wherein: ρ_(m1)=density of a firstmaterial; and ρ_(m2)=density of a second material. and wherein a secondactuator is controlled by a second signal based on a function f₂, where:f ₂=[(commanded dv _(in) /dt)×ρ_(m1)−commanded dm _(in)/dt]/(ρ_(m1)−ρ_(m2)).
 33. The control system of claim 31, wherein thecommanded dv_(in)/dt is based, at least in part, on a volumetric flowrate command feed forward input [dv_(s)/dt input] and the commandeddm_(in)/dt is based, at least in part, on an indication of density ofthe mixture and a first commanded density of the mixture.
 34. Thecontrol system of claim 33, wherein the indication of density isprovided by a density observer and the commanded dm_(in)/dt is furtherbased, at least in part, on a mass flow rate disturbance estimate outputby the density observer.
 35. The control system of claim 33, wherein thecommanded dv_(in)/dt is further based, at least in part, on anindication of height of the mixture in the tank and a commanded heightof the mixture in the tank.
 36. The control system of claim 35, whereinthe indication of height is provided by a height observer and thecommanded dv_(in)/dt is further based, at least in part, on a volumetricdisturbance estimate output by the height observer.
 37. A control systemfor mixing at least two materials, comprising: a controller operable tocontrol a physical system comprising a plurality of actuators, eachactuator being operable to introduce a material into a first tank toform a first mixture; wherein the controller operates the actuators tocontrol a desired characteristic of the first mixture.
 38. The controlsystem of claim 37, wherein the actuators are selected from the groupconsisting of valves, screw feeders, augurs, and elevators.
 39. Thecontrol system of claim 37, wherein the materials are selected from thegroup consisting of water, carrier fluid, dry cement material, sand, andproppants.
 40. The control system of claim 37, wherein the controllerfurther operates the actuators to control a desired operating parameterof the physical system.
 41. The control system of claim 37, wherein thecontroller further operates the actuators based on a combination ofsensors, the sensors selected from the group consisting of: a densitysensor measuring the density of the first mixture, a flow rate sensor,and a mixture height sensor measuring the height of the first mixture.42. The control system of claim 41, wherein the mixture height sensorcomprises a height observer.
 43. The control system of claim 41, whereinthe density sensor comprises a density observer.
 44. The control systemof claim 41, wherein the first mixture overflows the first tank into asecond tank to form a second mixture, and wherein the controller furtheroperates the actuators to control a desired characteristic of the secondmixture.
 45. The control system of claim 44, wherein the controlleremploys feed forward based on an input selected from the groupconsisting of: a combined inflow volume of the materials introduced bythe actuators, a volume of the first mixture flowing into the secondtank, a volume of the second mixture flowing out of the second tank, adensity of the first mixture, and a density of the second mixture. 46.The control system of claim 44, wherein the sensors are further selectedfrom the group consisting of: a density sensor measuring the density ofthe second mixture and a mixture height sensor measuring the height ofthe second mixture.
 47. The control system of claim 46, wherein themixture height sensor measuring the height of the second mixturecomprises a height observer.
 48. The control system of claim 46, whereinthe density sensor measuring the density of the second mixture comprisesa density observer.
 49. A method for mixing at least two materials,comprising: introducing the at least two materials into a physicalsystem; combining the at least two materials to form a mixture thereof;independently controlling a desired characteristic of the mixture of theat least two materials and a desired parameter of the physical system.50. The method of claim 49, wherein the controlling involves decouplinga volumetric flow rate disturbance parameter.
 51. The method of claim49, wherein the controlling involves decoupling a mass flow ratedisturbance parameter.
 52. A method for mixing at least two materials,comprising: determining a commanded combined mass flow rate of the atleast two materials into a first tank [commanded dm_(in)/dt];determining a commanded combined volume flow rate of the at least twomaterials into the first tank [commanded dv_(in)/dt]; and independentlycontrolling, based on the commanded dm_(in)/dt and the commandeddv_(in)/dt, a density of a mixture of the at least two materials in thefirst tank [ρ₁₂]and a volume flow rate of the mixture out of the firsttank [dv₁₂/dt].
 53. The method of claim 52, wherein the controllingincludes regulating at least two actuators, each actuator being operableto dispense one of the at least two materials, respectively, into thefirst tank.
 54. The method of claim 52, further including: determining acommanded mass flow rate of the mixture from the first tank into asecond tank [commanded dm₁₂/dt]; and wherein determining the commandeddm_(in)/dt is based on an indication of the density of the mixture inthe first tank and the commanded dm₁₂/dt; and determining a commandedvolume flow rate of the mixture from the first tank into the second tank[commanded dv₁₂/dt]; and wherein determining the commanded dv_(in)/dt isbased on an indication of a height of the mixture in the first tank andthe commanded dv₁₂/dt.
 55. The method of claim 54, further comprising:determining a desired height of the mixture in the second tank [inputh₂]; determining a desired volume flow rate of the mixture out of thesecond tank [input dv_(s)/dt], wherein determining the commanded dv₁₂/dtis based on the input h₂, the input dv_(s)/dt, and an indication of aheight of the mixture in the second tank; and determining a desireddensity of the mixture in the second tank [input ρ_(s)], whereindetermining the commanded dm₁₂/dt is based on the input ρ_(s) and anindication of a density of the mixture in the second tank.
 56. Themethod of claim 55, further comprising: sensing the density of themixture in the first tank [ρ₁₂ sensor]; determining an estimated densityof the mixture in the first tank [estimated ρ₁₂] based on the ρ₁₂ sensorand on the commanded dm₁₂/dt; providing the estimated ρ₁₂ as theindication of the density of the mixture in the first tank; sensing thedensity of the mixture in the second tank [ρ_(s) sensor]; determining anestimated density of the mixture in the second tank [estimated ρ_(s)]based on ρ_(s) sensor and on the indication of the density of themixture in the first tank; and providing the estimated ρ_(s) as theindication of the density of the mixture in the second tank.
 57. Themethod of claim 55, further comprising: sensing the height of themixture in the first tank [h₁ sensor]; determining an estimated heightof the mixture in the first tank [h₁ estimated] based on h₁ sensor andon the commanded dv₁₂/dt; providing the estimated h₁ as the indicationof the height of the mixture in the first tank; sensing the height ofthe mixture in the second tank [h₂ sensor]; determining an estimatedheight of the mixture in the second tank [h₂ estimated] based on h₂sensor and on the indication of the height of the mixture in the firsttank; and providing the estimated h₂ as the indication of the height ofthe second mixture in the second tank.